**1. Introduction**

Hyperloop is an innovative transportation system first outlined in a 2013 white paper by a joint team from Tesla Inc. and SpaceX Corp. As described in the Hyperloop Alpha document, it is a new, alternative, very high-speed mode of transport with benefits in terms of comfort, convenience, time, and cost [1,2]. The Hyperloop system consists of Hyperloop pods (or capsules) transporting cargo and passengers at very high speeds in a near-vacuum tube. The basic concept of the Hyperloop is similar to the evacuated tube transportation system proposed by Oster [3] in 1977. Estimates sugges<sup>t</sup> that the trip between Los Angeles and San Francisco of roughly 350 miles, which usually takes 2.5 h by a high-speed train, would take only 35 min by the Hyperloop system (estimated time at average speeds of approximately 1000 km/h and 240 km/h for Hyperloop pod and high-speed train, respectively) [1,4]. This high-speed pod–tube configuration could be faster and more energy-efficient than trains or cars, as well as cheaper and less polluting than aircraft. SpaceX founder Elon Musk and his team called it the fifth form of public transportation after planes, trains, cars, and boats [1].

There are two big differences between Hyperloop and traditional rail. First, the pods are designed to float on air bearings or by magnetic levitation to reduce friction, unlike trains or cars. Second, the pods (or capsules) transport passengers through tubes in which most of the air is evacuated to reduce air resistance. This should allow the pods to travel at approximately 1250 km/h (or nearly 350 m/s), which is 3–4 times higher than the fastest high-speed train at present (350–380 km/h). Such high-speed operation, however, would cause a larger aerodynamic drag. Addressing this challenge, the closed partial-vacuum tube could drastically lower the aerodynamic drag. Hence, the Hyperloop white paper proposed maintaining a constant pressure of 1/1000 atm (101.325 Pa) inside the tube. In addition, when the pod is operating at transonic speeds, the restricted air owing to air compressibility could lead to large variations in pressure at the front and rear of the pod. Hence, variation of pressure wave propagation should also be considered.

Oh et al. [5] conducted a large parametric study in steady state by applying a two-dimensional axisymmetric model to a compressible flow to investigate the e ffects of blockage ratio (BR), pod speed/length, and tube pressure/temperature on the aerodynamic drag in a Hyperloop system. The study suggested that these parameters strongly a ffected drag, except the pod length and tube temperature, which have negligible influences. However, the simulation in their study could not analyze pressure wave propagation. In steady-state simulation, the pressure waves produced by the nose and tail of the pod a ffect the inlet and outlet boundaries. This limitation can influence the accuracy of the results. Yang et al. [6] concluded that aerodynamic drag increased proportionally with the internal tube pressure and the square of the operating speed. Gillani et al. [7], Singh et al. (2019) [8], and Choi et al. [9] studied the relationship between the pod shape and the aerodynamic drag of the pod. They proposed that the elliptical train shape e fficiently reduced the aerodynamic drag at the tube pressure of 1013.25 Pa. Zhang [10] carried out steady computational fluid dynamic (CFD) simulations and indicated that the increase of BR increased the aerodynamic drag. Similar conclusions were presented by Kang et al. [11], who performed parametric simulations of transonic vehicles in an evacuated tube. The authors implied that the BR and internal tube pressure strongly a ffected aerodynamic drag of transonic trains; the maximized drag coe fficient was obtained at Mach 0.7. Kim et al. [12] concluded that the occurrence of shockwaves greatly increased the aerodynamic drag in the tube–train system.

The compressibility of air should be considered in cases where a pod travels through a tube at high speed. The motion of the pod generates a series of compression waves in the front. These pressure waves propagate forward and backward with the direction of the operating pod. Meanwhile, expansion waves propagate behind the pod. The propagation of compression waves intensifies the pressure ahead of the pod, whereas the expansion waves reduce the pressure behind the pod. The di fference between the pressures at the nose and the rear of the pod increases as the pod speed increases, causing a sharp rise in pressure drag. A study by Oh et al. [5] observed choked flow in the Hyperloop system at a pod speed of 180 m/s and a BR of 0.36, which significantly increased the drag. The normal shockwave is similar to a step-change of the compression wave. The presence of shockwaves produces discontinuous changes in the flow parameters. The normal and oblique shockwaves in the rear of the pod strongly interfere with the aerodynamic characteristics of the pod–tube system. Hence, in this study, the pressure wave propagation speed is also evaluated.

The analysis of pressure wave propagation requires time-dependent CFD simulation. In addition, a moving overset mesh method was employed to represent the pod motion and determine the influence of a moving pod on the compression waves and expansion waves inside the tube. The overset mesh is a renowned method for moving-mesh simulations and has high accuracy and computational speed by reducing the re-meshing e ffects during the simulation.

Oh et al. showed that symmetric models of a semicircular nose and tail make the di fference between three- and two-dimensional models insignificant (just 4%) [5]. Therefore, the two-dimensional axisymmetric model is su fficient for simulating the Hyperloop with an idealized geometrical shape. Therefore, in this study, a two-dimensional axisymmetric model was constructed. Moreover, the main objective of this study is to analyze the aerodynamic drag and pressure wave propagation, which can be fully described by two-dimensional simulations. The low computational cost of the two-dimensional model allowed us to consider a long tube and carry out a comprehensive parametric study.
